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Collective action and network change Takacs, Karoly; Janky, Bela; Flache, Andreas

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DOI: 10.1016/j.socnet.2008.02.003

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Social Networks

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Collective action and network change

Karoly´ Takacs´ a,b,∗,Bela´ Janky c, Andreas Flache b a Corvinus University of Budapest, Institute of Sociology and Social Policy, Hungary b University of Groningen, Faculty of Behavioral and Social Sciences, Department of Sociology/ICS, The Netherlands c Budapest University of Technology and Economics, Department of Sociology and Communication, Hungary article info abstract

Keywords: Network models of collective action commonly assume fixed social networks in which ties influence Collective action participation through social rewards. This implies that only certain ties are beneficial from the view Social dilemmas of individual actors. Accordingly, in this study we allow that actors strategically revise their relations. Social networks Moreover, in our model actors also take into account possible network consequences in their participation Network dynamics choices. To handle the interrelatedness of networks and participation, we introduce new equilibrium Social control Structural balance concepts. Our equilibrium analysis suggests that structures that tend to segregate contributors from free Local interaction games riders are stable, but costless network change only promotes all-or-nothing participation and complete networks. © 2008 Elsevier B.V. All rights reserved.

1. Introduction work members have strong incentives to insulate themselves from pressures to contribute. Based on different behavioral assumptions, Why and under which social conditions are groups successful in other theoretical studies supported the view that dense networks mobilizing collective action? Voluntary participation in collective may sometimes undermine rather than facilitate the enforcement actions, such as fund raising, strike movements or political upris- of contribution (Flache and Macy, 1996; Flache, 1996, 2002; Kitts et ing, seems often to contradict the narrowly defined self-interest al., 1999). The theoretical argument focuses on the desire of actors of the participants. Yet, empirical examples of successful mobiliza- to obtain social rewards from other group members including those tion abound. Students of collective action point to social networks who “free ride”. The desire to retain relationships with or attain as an important answer (see, e.g., Oberschall, 1973; Tilly, 1978; behavioral confirmation from free riders may often compromise Oliver, 1984; McAdam, 1986; Marwell et al., 1988; Gould, 1993a; actors’ willingness to exert social control towards contribution, Sandell and Stern, 1998; Chwe, 1999, 2000; and for an overview especially in a closely knit network. Diani, 2003a). Dense networks of communication and interac- To clarify under what conditions networks are positively or neg- tion between prospective participants may greatly facilitate group atively related to collective action success, models of collective mobilization (Opp and Gern, 1993; Gould, 1993b; Marwell and action need to incorporate explicitly how individual actors make Oliver, 1993). The view that social networks facilitate collective purposive decisions to use their social relations to foster their goals, action relies on the assumption that individual network mem- be it to enforce compliance or to resist peer pressure. But studies bers have a “regulatory interest” (Heckathorn, 1988; Kitts, 2006) that combine positive and negative effects of social ties on cooper- to enforce others’ contribution to the collective action. Particularly ation (e.g., Oberschall, 1994; Heckathorn, 1996; Takacs,´ 2001)have in dense or closed networks, actors can effectively employ their neglected a crucial implication of this perspective. Purposive action social ties for this purpose (Hechter, 1987; Coleman, 1990)both implies that network members in a collective action situation not because group members have more information about one another only use existing ties to attain their goals, but they may also make and because they have more social means to provide rewards for or break ties if this serves their purposes. compliance or punish deviance. In general, most models of collective action that address social But it has also been argued that network ties have a “double network effects implicitly assume that there is a fixed set of edge” (Flache, 1996). Heckathorn (1996) has pointed out how peer interpersonal relations that do not change over time. Relational pressure may take the form of “oppositional control”, when net- ties are in these studies exogenously given and at most, static comparisons are made. We argue that the relationship between collective action and social networks cannot be properly stud- ∗ ied without addressing endogenous network change driven by Corresponding author at: Corvinus University of Budapest, Institute of Sociology and Social Policy, H-1093 Budapest, Kozrakt¨ ar´ u. 4–6, Hungary. individual interests. Regulatory interests do not necessarily lead E-mail address: [email protected] (K. Takacs).´ to enforcement, but possibly also to avoidance of unpleasant

0378-8733/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.socnet.2008.02.003 178 K. Takacs´ et al. / Social Networks 30 (2008) 177–189 control and optimization of contacts. For illustration, consider the understand collective action and stability of social networks. The situation of a defector who is particularly sensitive to conformity main added value of this study for the literature on network for- pressure and who has her ties mostly with compliant group mem- mation is the development and application of such equilibrium bers. In a static network, it is likely that pressure to contribute concepts to n-person games. brings this defector back into line. But if networks can change, the Another line of work explicitly combines collective action with defector faces an incentive to maximize the number of ties she has network change (Kitts et al., 1999), but uses “backward-looking” with other defectors and thus break ties with cooperators and build models of structural learning that describe the process by which new ties with other defectors. If the latter mechanism prevails, individuals strengthen ties that are beneficial for them and aban- the outcome on the collective level may be a disconnected net- don ties with negative experience (Macy et al., 2003). These models work with a deviant clique on which cooperators cannot effectively do not assume that decision making is strategic and purposive in impose peer pressure. But if the first mechanism is more important, the sense that individual actors weigh the costs and benefits of the result may be a network in which free riders are effectively changing their contribution behavior against the costs and benefits sanctioned by their compliant peers. In this example, models of of making or breaking relationships, or a combination of both. This collective action that neglect endogenous network changes might makes it difficult to address effects of exogenous constraints, such arrive at questionable conclusions. as given access structures of networks or communication costs, In this study we emphasize that networks change over time and within the structural learning framework. that this is also reflected in how people behave in collective action In this paper, we propose a game theoretical model of collective (Kim and Bearman, 1997; Diani, 2003b; Osa, 2003; Gould, 2003). action in dynamic networks and arrive at equilibrium predictions. Besides being driven by independent network dynamics, networks Our main innovation is that we incorporate into a collective action may also change because of the internal determinants of collective framework the possibility of tie formation and deletion, and model action. People might choose their structural relations strategically these decisions as interdependent with participation decisions. We in order to maximize rewards and minimize punishments that orig- draw on game theoretic studies of network evolution and model inate in social control (cf. Harsanyi, 1969). Furthermore, in order both collective action and relational change as the result of strate- to facilitate participation, strategic establishment of communica- gic and purposive decision making that explicitly reflects the costs tion channels and other linkages might take place (McAdam and and benefits of the various decision options involved. The game Paulsen, 1993; Diani, 2003c). Strategic tie formation goes very theoretic approach allows us to arrive at equilibrium predictions far in the proliferation of social movement politics. Campaign- without the need of using simulation techniques. We will use ing and lobbying often involves tie-formation strategies, popularly these equilibrium predictions to identify exogenous conditions that called as “networking” (e.g., Tilly and Wood, 2003). On the other explain why sometimes network dynamics may undercut the effec- hand, another branch of literature emphasizes individual opportu- tiveness of peer pressure, while network influences successfully nities that arise from deleting relations and from structural holes foster collective action at other times. Examples of phenomena our (e.g., Burt, 1992, 2005; Buskens and van de Rijt, 2005; Burger model can tackle are the formation of deviant cliques that remain and Buskens, 2006). Severing links is also considered as a pur- stable despite strong social control, or work teams where team- poseful action in different lines of social psychological literature segregation develops along a division line between workers and (see Williams et al., 2005 for an overview). Studies show that in shirkers, even when high team performance would be in every- collective action, ostracism and the threat of exclusion increases body’s interest. participation (Olson, 1982; Hirshleifer and Rasmusen, 1989; Ule, Our work is a first step towards addressing such phenomena in a 2005; Ouwerkerk et al., 2005). framework that integrates collective action dynamics and network There is also empirical evidence on how collective action might dynamics with the assumption of individual strategic behavior and restructure individual relations. Activism has certainly a role in the notion that network ties are conduits of social rewards in col- changing the meaning and impact of interpersonal ties (Gould, lective action. Accordingly, the major contribution of the paper will 2003). As a result of the dynamics of dyadic relations, macro prop- be theoretical. To provide the tools to tackle the substantive ques- erties of the network and thus perspectives of (further) collective tions we are interested in, we develop an equilibrium concept that action also change. Our contribution along this line is to inte- embraces Nash equilibrium in the n-person collective action game grate the perspective of network change and the resulting network and network stability at the same time. The concept, called strongly dynamics in a theoretical model of collective action. robust network equilibrium, delineates structural conditions under Independently from the literature we cited so far, considerable which group outcomes are stable both in terms of the network and progress has been made recently in modeling the dynamics of in terms of the collective action decisions of the group’s members. social networks as a result of strategic individual action (Jackson In the remainder of the paper, we proceed as follows. In Section and Watts, 2002a; Dutta and Jackson, 2003; Ule, 2005). Previous 2, the basic model of social control and collective action is lined out. game theoretic models, however, concentrated on strategic net- In Section 3, conditions are derived under which strategic actors work formation without collective interdependencies (Bala and will not delete any tie or form new ones in a given strategy profile Goyal, 2000; Bonacich, 2001). Most studies use games of strate- of collective action decisions. Section 4 contains a derivation of the gic network formation that assume values for established ties in conditions for strongly robust network equilibria. Finally, we dis- cooperative situations (Dutta et al., 1998; Slikker and van den cuss our results in Section 5, where we also formulate conclusions Nouweland, 2001) or indirect benefits from the chain or network of and suggestions for further research. contacts (Jackson and Wolinsky, 1996; Jackson and Watts, 2002b). In this literature on network formation, equilibrium concepts have been developed that describe equilibrium networks in which no 2. The model of social control and collective action individuals have incentives to delete existing ties or build new ones (Jackson and Wolinsky, 1996; Watts, 2001; Gilles and Sarangi, This section provides a description of our model of decisions on 2004; Ule, 2005; Buskens and van de Rijt, 2005; van de Rijt and participation in collective action. Collective action is modeled as Buskens, 2005; Goyal and Vega-Redondo, 2007). If network stabil- a one-shot n-person public goods game with a linear production ity and behavioral stability are embraced and adjusted to collective function and binary decisions and it is integrated with a model of interdependencies, similar concepts could help us to model and dyadic interdependence between directly related players. In Sec- K. Takacs´ et al. / Social Networks 30 (2008) 177–189 179 tion 3, we extend this model by a network game, in which players be assured that many friends choose to stay away from the demon- can change interpersonal connections. stration. As effective control from more contacts pushes stronger We define N to be the set of actors, where N contains n (n >2) towards participation, individuals face “local thresholds” of partic- players.1 Individual actions are binary and summarized in vector ipation similar to the impact of a critical mass of participants on ␴, in which the action of i is denoted by i, where i = 1 is con- individual participation in other models of collective action (Macy, tribution and i = 0 is non-contribution or defection (for all i ∈ N). 1991; Marwell and Oliver, 1993; Chwe, 1999). We use (i, −i) to refer to the action profile chosen by all group Second, behavioral confirmation is also dependent on the pro- members, where we distinguish between i’s action and the actions portion of contacts with identical action (from all friends of i). We ∈ chosen by other members of the group, −i. Each participation (con- assume that the subjective payoffs i for all i N increase by b2/ri for tribution) provides a unit of public good ˛ for all individuals. This each j if i and j have a network tie (ij ∈ R) and their actions are identi- means that the more people participate, the higher the public good cal (i = j). The parameter b2 scales the strength of this proportional reward each player receives. Participation has an individual cost c, conformity effect. Different lines of arguments justify this assump- which is higher than one unit of public good (c > ˛), but smaller tion. In-degree and the proportion of relevant others behaving the than n units (c < n˛). These payoff restrictions impose a Prisoner’s same way (e.g., mean friendship delinquency) are important pre- Dilemma structure in the public goods game. When we neglect dictors of peer influence (Haynie, 2001). Furthermore, a study using social network payoffs, universal contribution is Pareto superior to working unit-level data found that the proportion of participants in universal defection, but unilateral defection is individually rational. the working unit is a significant predictor of individual strike partic- Actors play this game not in isolation. They might have con- ipation (Dixon and Roscigno, 2003). Including a proportional term nections (links) to each other, which define a social network. For in the model is also in line with theories on cultural change that the sake of simplicity, we consider undirected and unvalued con- emphasize that individuals are influenced by relatively more fre- nections, which implies that every link is symmetric and equally quent cultural traits around them (e.g., Boyd and Richerson, 1985) important. The social network (N, R) is characterized by the set of and with network models of coordination (Buskens and Snijders, actors N, and by the set of connections R ⊂ RN, where RN = {ij |{i, 2005). j} ∈ N × N, i = j} and ij = ji. Sometimes we will refer to j as i’s con- In addition, structural relations are the sources of positive social tact or friend if ij ∈ R. As we consider a fixed network size, we will selective incentives (s) that reward contribution. We assume that simply refer to the social network (N, R)asR and to the network the subjective payoffs i for all i ∈ N increase by i s from each ∈ from which the relation ij is deleted as R\{ij}. The set Ri = {ij|ij ∈ R, ij R connection, hence by isri in total. This implies that only j ∈ N, i = j} ⊂ R contains all ties of i and the set Rˇ i = {ij|ij ∈/ R, j ∈ N, contributors receive selective incentive rewards. If the provision of i = j} denotes the null-dyads of i. The size of Ri is the number of ties selective incentives is costly, actors would not only face a public (degree) i has and is denoted by ri. Actor i is an isolate if ri =0. goods problem, but also a second order free-rider problem (Oliver, If ij ∈ R, actors i and j influence each other. First, we assume that 1980; Heckathorn, 1989). As we concentrate, however, on selective actors prefer to act the same way as their contacts (cf. Chong, 1991; incentives of a social character like respect or status, we deem Oberschall, 1994), which we will refer to as behavioral confirmation. it plausible that these are produced without costs (Chong, 1991; This assumption is supported by empirical studies that show that Lovaglia et al., 2003) and that therefore no second order free-rider individuals even gather information on-the-spot to assess whether problem arises (see also Coleman, 1990). This assumption receives their friends will participate or not (Oberschall, 1993; Dixon and additional support by evidence that individuals punish defectors Roscigno, 2003). When the actions of contacts match, we assume voluntarily, in spite of sanctioning costs (Fehr and Gachter,¨ 2000, that they obtain a higher subjective payoff than when their action 2002; Boyd et al., 2003). is different. This kind of strategic interdependence is also modeled We use these assumptions to express the subjective payoff play- in coordination games played on a network (see, e.g., Berninghaus ers obtain given their choice for participation or defection. The sub- and Schwalbe, 1996; Morris, 2000; Chwe, 2000; Jackson and Watts, jective individual payoff i of i ∈ N will depend on the number of her 2002a; Buskens and Snijders, 2005). Our study, however, moves contacts who participate, ric (where ric is the size of Ric = {ij|ij ∈ Ri, beyond this framework and introduces strategic interdependence j =1} ⊂ Ri), and on the number of her contacts who do not partic- − ∈ not only in terms of the need of coordination, but also in terms ipate (rid = ri ric). Formally, i for all i N with ri > 0 in the given of the more challenging problem to achieve cooperation between network R and given strategy profile ␴(i, −i) is determined as: self-interested actors in an n-person Prisoner’s Dilemma. n We assume that behavioral confirmation might take two differ- = = + rid + i(i 0,−i, R) ridb1 b2 ˛ j ent forms. First, we assume that the subjective payoffs i for all ri j=1,j=i i ∈ N are increased by the value of mass conformity b1 for each j if ∈ ij R and i = j. This means that an additional contact has the same and influence on the actor as other contacts, and hence the number of ⎛ ⎞ contacts with identical choice determines the strength of this type n = = + + ric + ⎝ + ⎠ − of influence. Empirical research on social influence supports that i(i 1,−i, R) ris ricb1 b2 ˛ j 1 c, ri individuals are more likely to choose a certain behavior when the j=1,j=i number of relevant others who behave this way increases (see for instance, Ploeger, 1997 and Snijders and Baerveldt, 2003 for delin- which can be compactly written as quency; Blum et al., 2000 for alcohol use; Simons-Morton and Chen, b2 2006 for substance use). For example, when mass conformity oper- (␴, R) = ˛ + r s + (r − r ) b + − c i i ic id 1 r i ates, an individual, who intends to participate in a demonstration, i is pleased when there are many friends in the crowd. On the other n + + b2 + hand, in case this individual prefers to stay at home, she is pleased to rid b1 ˛ j, (1) ri j=1 ∈ \{ } ␴ = − + n where j N i .Ifri =0,i( , R) (˛ c)i ˛ = j. In the main 1 Sets will be denoted by bold capitals and their sizes by corresponding small j 1 letters in italics. For example, |N | = n. text, we assume that contributions have positive externalities, ˛ >0, 180 K. Takacs´ et al. / Social Networks 30 (2008) 177–189 and social control generates rewards (b1 ≥ 0, b2 ≥ 0, and s ≥ 0). Def- lyzing network equilibria, we will assume that a new tie is initiated inition of payoff functions if social control appears in terms of pun- only if it is part of an (extended) individual network that provides ishments and the analysis of such situations is left to Appendix B. higher benefits than the present network, taken into account the Now we can derive the conditions for individually rational par- costs of change and under the assumption of no other change in ticipation in collective action given a certain network position and the network. Besides this basic and simplified cost–benefit analy- given a vector of participation decisions of group members. Broadly, sis, following previous work, we will assume that consent of the an actor will participate if the subjective payoffs from contribu- other party is required to form a new connection. tion exceed the subjective payoffs from free riding in the given We are also concerned with the possibility that players might network and strategy profile. This is achieved when the selective benefit from the combination of deleting and building some ties. incentives and behavioral confirmation rewards she receives in Consider, for instance, a defector i who has only a single tie ij and this her present network position are sufficient to compensate for the leads to a contributor j. Since proportional conformity rewards in = ≥ costs of contribution. Formally, i participates if i(i 1,−i, R) this case are zero, player i has no incentive to break ij if a > 0. She also = i(i 0,−i, R) that is when has no incentive to be tied with other defectors when f > b1 + 0.5b2. Replacing ij with a connection to another free rider, however, is b ˛ + r s + (r − r ) b + 2 ≥ c. (2) beneficial for i in case b + b > f + a. Such a situation is not unlikely i ic id 1 r 1 2 i when the cost of deleting a relation is small. Expression (2) asserts that the effect of selective incentives on Most equilibrium formulations in the research on games of net- participation increase in the number of ties of the given individual. work formation,6 such as the stable network concept of Watts Behavioral confirmation promotes contribution only when more (2001) or pairwise stability of Jackson and Wolinsky (1996), posited contacts contribute than defect.2 two requirements (for a review, see Dutta and Jackson, 2003). The first requirement prescribes that no player intends to delete a con- 3. Network changes nection and the second requirement is that no player intends to add a new relation (Watts, 2001) or no new tie could be formed for As a next step, we include the possibility that actors might delete the mutual benefit of the players involved (Jackson and Wolinsky, their existing connections and might form new ones. For the time 1996). A stronger version of the latter concept is strong pairwise being, we treat participation decisions as fixed. This allows to iden- stability that combines pairwise stability and strong link deletion tify stable network structures at a given strategy profile of the public proofness by allowing multiple ties to be deleted, but only a single goods game.3 We continue to assume that social control parame- tie to be built at a time (Jackson and Wolinsky, 1996; Gilles and ters are rewards (b1 ≥ 0, b2 ≥ 0, and s ≥ 0). The costs of abandoning Sarangi, 2004). More complex stability concepts, such as strong one tie are denoted by a and the cost of forming one new tie is stability, allow for a coalition of players that is larger than two denoted by f (a ≥ 0, f ≥ 0). to deviate (see Dutta and Mutuswami, 1997; Jackson and van den A connection ij is not stable if i or j prefers to delete it.4 No Nouweland, 2005; Jackson, 2004). compensation payments are possible to save the connection (cf. For our purposes, we need a simple concept that is still non- Bala and Goyal, 2000). Note that it might be beneficial for a player cooperative in nature and unlike in strong pairwise stability, to delete a set of her links at once. Even if a deletion of ij does not considers multiple tie formation. The point of departure is based on result in a higher payoff, it can be part of the set of links that is the Nash logic and states that in equilibrium it should not be in any- rewarding to be deleted. one’s interest to make “any change” in her network. By “any change” On the other hand, we exclude the possibility of one-sided tie we mean any combination of deleting existing and of forming new formation, because we use non-directed relations (cf. Bala and relations in which the individual is involved. This stability concept Goyal, 2000; Buskens and van de Rijt, 2005). To keep our analy- is useful for the discussion of conditions under which a network sis simple, we abstract from the possibility that actors may bargain is free of tie deletions and initiations. Applied to collective action about which actor or which coalition of group members may bear situations, this stability concept would characterize networks in a the costs of the formation of some subset of ties. Tie formation is given strategy profile in the (collective action) game, if there is no assumed to have the same costs for both parties and no coalition actor i, for whom any change in the set of her relations would result possibilities beyond the dyad are taken into account.5 When ana- in a better outcome given that relational contacts outside of i are fixed. Formally, we would call R resistant to changes inagiven␴, if i(␴, R) ≥ i(␴,(R ∪ Gi)\Hi) − hia − gif for any Gi ⊂ Rˇ i,anyHi ⊂ Ri and for all i ∈ N, where the notations H are introduced for the set 2 A more refined analysis of this static model can be found in Janky and Takacs´ i (2005). Main results for the entire network show that the minimum degree of of ties to be deleted and Gi for the set of ties to be built by i ∈ N. The the network is a strong determinant of overall collective action in case selective sizes of these sets are denoted by hi and gi, respectively. incentives operate. Network clustering has a strong influence when behavioral con- It is not our main interest to deal with situations in which actors firmation mechanisms are strong and might undermine mass collective action. can impose ties on others (as in directed networks). We also do not Clustered networks are more likely to have partial contribution equilibria, in which participants and free riders are segregated. The smaller the number of free riders in want to focus on whether initiations of new ties take place, but we the partial contribution equilibrium, the less likely that full contribution is a payoff want to address whether new ties are built or not, which requires dominant equilibrium. Moreover, the payoff dominance of full contribution equilib- consent also from the other party. Accordingly, we need to move rium is not likely in centralized structures when mass conformity is strong, but it beyond the use of a purely Nash-based concept. As ties are sym- is possible in case proportional conformity is prevalent. A further interesting result metric and tie formation requires mutual consent, we build our is that not only behavioral confirmation, but also selective incentives might have a non-monotonic effect on the existence of full contribution equilibria. equilibrium concept on the same logic as monadic stability (Gilles 3 A version of the model allowing only for network decline is also discussed in and Sarangi, 2004) and unilateral stability (Buskens and van de Rijt, Takacs´ and Janky (2007). 4 When behavioral confirmation and social selective incentives are negative, more links are not stable than when behavioral confirmation and selective incentives are positive (see Appendix B). 1996; Dutta et al., 1998; Slikker and van den Nouweland, 2001; Jackson and van den 5 Bargaining could be incorporated with a coalition analysis of network forma- Nouweland, 2005) do not discuss collective action. tion including the use of equilibrium concepts from cooperative game theory, such 6 This line of research requires a characteristic function defined for the network as the core. However, studies that use this approach (e.g., Jackson and Wolinsky, and an allocation rule, but its basic definitions can also be applied to our case. K. Takacs´ et al. / Social Networks 30 (2008) 177–189 181

2005). These concepts take into account that a new tie is only tive rewards. The more ties to cooperators the defector breaks, the formed if it is to the benefit of the partner, as well. Monadic stability larger is the improvement of the ratio of defectors to cooperators in is based on the idea that players take it as granted that other players the defector’s personal network. This implies that any additional tie respond affirmatively to an initiation if the new link is profitable to abandoned yields a greater benefit than the previous one, whereas them, but no further consequences are taken into account (Gilles the costs are the same for each deletion. Technically, after substi- and Sarangi, 2004). Buskens and van de Rijt (2005) define a network tuting ric for hi, the necessary conditions of a beneficial structural unilaterally stable if no actor would be better off by changing her ties change for defector i with rid > 0 are given as or if anyone was, then at least one actor whom she proposes a new ric + − tie is worse off in the new network than in the original network. b2 >rica gi(f b1). (4) ri We will adopt this definition and adjust it for our purposes when ∈ applying to collective action problems and network formation. Similarly, for a contributor i N with ri > 0 a structural change in We define a social network unilaterally stable in a given strat- which she abandons hi ties to defectors and newly forms gi ties to egy profile in the (collective action) game, if there is no actor i,for contributors is beneficial, when whom any change in the set of her relations would result in a bet- g r + h r (g − h )s + g b + b i id i ic >ha + g f (5) i i i 1 2 − + i i ter outcome or if a change would result in a better outcome then ri(ri hi gi) it does not satisfy some of the new partners given that relational holds. From (5) it can be seen that for a contributor forming the contacts outside of the scope of i are fixed. This stability concept first new relation with another contributor is always at least as does not allow coalition formation beside the involved dyads, but profitable as further ones, even when considering the simultane- requires consent from new partners involved. ous possibility of deleting ties. Again, this is caused by proportional Definition. R is unilaterally stable inagiven␴, conformity. The less ties a cooperator has (but at least one to a defector), the larger is the improvement in the ratio of defectors to cooperators from ego’s point of view if an additional tie is estab- if i(␴, R) ≥ i(␴,(R ∪ Gi)\Hi) − hia − gif lished with a similarly acting group member. Hence, the first new or exists such j ∈ N that ij ∈ Gi: j(␴, R)>j(␴, R ∪ {ij}) − f tie is the most valuable for a cooperator, in case she has at least for any Gi ⊂ Rˇ i,anyHi ⊂ Ri and for all i ∈ N. one connection to a defector (or if she is an isolate). Technically, after substituting 1 for g , the necessary conditions of a beneficial From the first sight, next to considering a given strategy profile i structural change for contributor i with r > 0 are given as in the collective action game, there is a further difference compared id to the original formalization of unilateral stability. Here, new part- r + h r b + b id i ic >ha + f + (h − 1)s. (6) 1 2 − + i i ners compare subjective rewards from the original network with ri(ri hi 1) subjective rewards from the network to which their new tie with A network is unilaterally stable in a given strategy profile, if there i is added. Note that in our case this is just a simplified formaliza- is no defector for whom Eq. (4) is satisfied with any values of g and tion: as strategies are fixed and there are no indirect rewards from i there also is no contributor for whom Eq. (6) holds with any values network changes, (␴, R ∪ {ij})= (␴,(R ∪ G )\H ) for all such j ∈ N j j i i of h . Networks that contain ties between defectors and contribu- that ij ∈ G . i i tors are most likely to be unilaterally stable when costs of link dele- For the simple case where a = 0 and f = 0, there is a clear-cut tion (a) and link formation (f) are high and every individual is tied to answer to the question which networks can be stable. Only a dis- several others.7 Substantively, such conditions can be interpreted as connected network with a complete component of contributors and high constraints on network change imposed by the given network. a complete component of defectors can sustain unilateral stability. Besides these general results, we can formulate some illustra- The reason is that defectors always gain from abandoning all ties to tive statements that characterize equilibria. Theorem I summarizes contributors and contributors always gain from building as many three results concerning deleting a tie ij in a given strategy pro- as possible new ties to fellow contributors. On the other hand, if file ␴ (see Appendix A for the proof). A tie ij ∈ R will be called selective incentives are large enough relative to proportional con- i not stable if there exist G ⊂ Rˇ , and H ⊂ R such that ij ∈ H and formity, then even a disconnected network might not be resistant to i i i i i (␴,(R ∪ G )\H ) − h a − g f > (␴, R). changes, because cooperators would prefer to connect to defectors i i i i i i as well. We do not obtain such straightforward results, however, Theorem I. In any R and given ␴, for all i,j ∈ N and ij ∈ R: if there are costs of deleting and forming ties. In these cases, net- works that contain ties between defectors and contributors can also (a) If i = j, then i(␴, R)≥ i(␴, R\{ij}). be unilaterally stable. (b) If i =0,j =1,rid >0,and b2/ri > a, then ij is not stable. A contributor might profit from a new tie to a defector, but such (c) If i =0, j =1, b2 >0 and for sufficiently small a and f, ij is not a tie will never be beneficial for the defector. As consent from both stable. parties is required for a new relationship, such a tie will not be ∈ realized. In general, a defector i N with rid > 0 is better off by a Part (a) of Theorem I claims that no player can increase her sub- structural change in which she deletes hi ties to contributors and jective payoff by deleting a link to another player who acts the same newly forms gi ties to defectors, if way as she does in any network and given strategy profile. Part (b) states that if there is a tie that connects a defector and a contrib- g r + h r g b + b i ic i id >ha + g f (3) utor, if the defector has at least one tie to another defector and i 1 2 − + i i ri(ri hi gi) holds. This condition implies that for a defector it is always more profitable to abandon all relations with contributors rather than 7 In case only negative social control operates, calculating which networks are just breaking up some of them, even when tie formation is simulta- unilaterally stable is easier. In Appendix B we show that in this condition no com- neously possible. The reason is that for a free rider who has at least bination of deleting and building ties can be individually profitable. An individual either has an incentive to delete all her ties or has an incentive to form a new one. one contact to another defector breaking contacts to actors with Hence, a strong link deletion proof network (see formal definition later) in which dissimilar strategies improves proportional conformity rewards no dyad is interested to form a new connection will be resistant to changes (and and does not yield any loss of mass conformity or selective incen- unilaterally stable). 182 K. Takacs´ et al. / Social Networks 30 (2008) 177–189 costs of tie deletion are under a threshold that is determined by proportional conformity and the individual degree of the defec- tor, then this tie is not stable (at least the defector wants to delete this tie). Furthermore, part (c) asserts that relations between defec- tors and contributors are not stable, if proportional confirmation is positive and tie deletion and formation costs are under a specific threshold. To analyze macro-level consequences of Theorem I,we introduce stability concepts for the entire network that concern link deletions:

Definition. R is link deletion proof inagiven␴,ifi(␴, R) ≥ i(␴, R\{ij}) − a for all i ∈ N and ij ∈ R.

That is, a social network is link deletion proof in a given strat- egy profile (in the collective action game) if there is no actor for whom deleting a single relation would result in higher subjective payoffs, assuming exactly the same actions and no other change in the network. This concept of link deletion proofness, similar to stable networks (Watts, 2001) and pairwise stability (Jackson and Fig. 1. Segregation of contributors and link deletion proofness. Notes: Example for Wolinsky, 1996), concerns only a single change in the network at parameter values: s =1,b1 =1,b2 =5,c − ˛ = 3.5, a = 2. Filled nodes denote contributors once. A stability concept that allows players to delete any set of links and empty nodes are defectors. (a) A less segregated network and link deletions. (b) at once is called strong link deletion proofness (Gilles and Sarangi, A more segregated network and link deletion proofness. 2004; Belleflamme and Bloch, 2004). the highest degree among defectors who have connections to con- Definition. R is strong link deletion proof inagiven␴,ifi(␴, tributors is decisive for strong link deletion proofness of structures R) ≥ i(␴, R\Hi) − hia for any Hi ⊂ Ri and for all i ∈ N. with mixed ego-networks. The higher the maximum degree within this subset, the more likely it is that a structure with mixed ego- That is, a social network is strong link deletion proof in a given networks can also be strong link deletion proof. This implies that strategy profile (in the collective action game) if there is no actor for highly centralized and very dense networks that contain individ- whom deleting any subset of her relations would result in higher uals with a high degree are more likely to be strong link deletion subjective payoffs, assuming exactly the same actions and no other proof.9 change in the network. Adopting the concept of strong link deletion Corollary I.b shows that if costs of tie deletion are low, a discon- proofness, we can form two corollaries of Theorem I. nected network in which defectors are only tied to defectors and Corollary I.a. Any R is strong link deletion proof if ␴ = 0 or ␴ = 1. contributors are only tied to contributors will be strong link dele- tion proof. One should note, however, that a bipartite network, in ≥ Corollary I.b. If b2/rdmax > a 0(where rdmax is the highest individ- which all defectors are only tied to contributors and contributors ual degree among defectors), then in a strong link deletion proof R for are only tied to defectors will also be link deletion proof in this case. ∈ ∈ all i N with i =0holds that if ij, ik Ri then j = k. The level of segregation of contributors and defectors, however, The corollaries summarize typical cases under which a network does not have an unambiguous impact on link deletion proofness. is strong link deletion proof. Corollary I.a states that every net- If the network is perfectly segregated, then it has a component of work is strong link deletion proof in a full contribution and in a defectors and a component of contributors and hence it is strong full defection strategy profile. Corollary I.b claims that if propor- link deletion proof. But this does not imply that for those networks tional conformity rewards are positive and costs of tie deletion are that contain ties between contributors and defectors, a highly under a threshold value, then in a strong link deletion proof net- segregated network is necessarily prone to become even more seg- work in the collective action game every defector has ties only to regated, or a moderately segregated network is more likely to be defectors or only to contributors, but not to both. stable than a highly segregated one. The examples in Fig. 1 illustrate The key parameter that underlies Theorem I and the corollaries how a less segregated network (Fig. 1a) can be more subject to link deletion than a more segregated one (Fig. 1b). The two networks is proportional conformity (b2). Proportional conformity is respon- sible for the result that deleting all relations to contributors is more are identical concerning the sets of contributors and defectors and beneficial for a defector than just deleting one or some of them. have the same density. The numbers of connections of D1 and D3 The more links to contributors a defector deletes, the higher is the influence network changes in the less segregated network and only improvement of the ratio of defectors to contributors among her D1’s connections matter in Fig. 1b. The improvement of network ties. For example, if this ratio is 5:10, then deletion of one link composition in terms of proportional conformity is smaller for D1 improves the ratio by about 0.0556, while deletion of two links in network 1b than it is in network 1a. Hence, at a wide range to contributors yields an improvement of about 0.125, deletion of of parameter values, only the latter network is (strong) link dele- three yields an improvement of 0.214, etc. tion proof. Consequently, the initially less segregated network (in The threshold conditions in Corollary I.b for the costs of tie Fig. 1a) becomes more segregated after individual structural deci- deletion a illustrate that a high level of proportional conformity sions. The driving mechanism is again proportional conformity. makes networks in which defectors are tied both to contributors Bridging actors with many ties do not benefit as much in terms and defectors subject to link deletion.8 The conditions imply that of proportional conformity from deleting links to individuals with dissimilar choices, as compared to the benefits that less integrated bridging actors can obtain from deleting such ties.

8 In case of negative social control, mass conformity and selective incentives also play a role in strong link deletion proofness. Furthermore, contributors might also 9 The sign of the social control parameters does not alter our main conclusions, have an incentive to delete all their ties to free riders. Hence, there are stricter although the maximum degree among contributors with connections to defectors requirements for strong link deletion proofness—see Appendix B. is also relevant for strong link deletion proofness (see Appendix B). K. Takacs´ et al. / Social Networks 30 (2008) 177–189 183

We now turn to some equilibrium properties that concern tie link deletion, however, the level of segregation between contribu- formation. The main results concerning forming ties are summa- tors and defectors has a more clear-cut effect on network changes: rized in Theorem II (see Appendix A for the proof). A tie ij ∈/ R more segregated networks are less likely to grow. Nevertheless, the is initiated by i if exist Gi ⊂ Rˇ i, and Hi ⊂ Ri such that ij ∈ Gi and impact of individual degree on changes is stronger than the impact i(␴,(R ∪ Gi)\Hi) − hia − gif > i(␴, R). of the number of abridging individual connections. To sum up, our analysis of tie deletion and link creation has Theorem II. In any R and given ␴, for all i,j ∈ N and ij ∈/ R: suggested that relations in collective action tend to build up slowly and break up easily. We found that the most profitable strategy for a ␴ ≥ ␴ ∪ { } (a) If i =0and j =1,then i( , R) i( , R ij ). defector is to abandon all of her ties to contributors (assuming that ␴ ≥ ␴ ∪ { } − ␴ ≥ ␴ (b) If i = j and i( , R) i( , R ij ) f, then i( , R) i( , she prefers to delete any tie and has at least one relation to another ∪ − ∈ ⊂ ˇ R Gi) gif, where ij Gi for any Gi Ri. defector). On the other hand, we also showed that forming the first + + 2 + (c) If i = j =1and s b1 b2rid/(ri ri) >f, then ij is initiated by new tie to another defector has the highest marginal benefits, in i. case there is at least one connection to a contributor. + 2 + (d) If i = j =0and b1 b2ric/(ri ri) >f, then ij is initiated by i. For illustration, we highlight the main predictions of our model with network changes with a stylized example. Consider a wild Part (a) of Theorem II states that new relationships are not cat strike in a factory with a dense but not complete network of formed between contributors and defectors. The reason is that the informal social ties among workers. The strike can be modeled as defector does not gain anything from a new tie to a contributor. a one-shot public good game in which only informal social control Part (b) follows from that the marginal benefits of forming more fosters participation. When only a few workers participate, tensions ties are decreasing in the number of ties. Again, this is caused by between strikers and goons may emerge afterwards and might even proportional conformity. Hence, the first new tie is the most valu- result in breaking old relations. The model assumes that strikers able, in case there is at least one connection to a defector. If a single who are only related to strikers (goons only related to goons) get new tie is not beneficial, then no larger new set that contains this positive feedback from their peers, and their relationships do not tie can be feasible. Similarly, assuming a coalition of defectors in come under pressure by the event. Those who have a contact from which multiple ties are formed, the marginal benefits of forming the opposite camp, however, feel shame, guilt or are simply embar- new ties are decreasing.10 rassed by the conflict with some of their other contacts, which Parts (c) and (d) yield implications for the characteristics of those generates an incentive to break the relationship. Our model implies network positions in which actors are most likely to form new ties. that the ‘clearing’ of such a ‘mixed’ ego-network is more difficult To begin with, the number of defector contacts increases the chance to the extent that ego has many connections. Hence, our analysis that a new tie is formed between two contributors. Furthermore, suggests that a more dense community is more likely to remain contributors with many connections (high ri) are less likely to form cohesive even after the heated times of the wild cat strike. Where new connections to other contributors, as it does not give them workers are less embedded, however, contacts between strikers sufficient marginal benefits (if rid > 0). Considering two defectors and goons might dissolve and segregated fractions may be formed. that are not tied with each other, the likelihood of a new connec- The model also predicts that in a loosely tied group of workers, tion increases with the number of relations to contributors. Again, the collective experience of the demonstration might bring strik- individuals with many connections are less likely to form new ties ers closer to each other. Nonetheless, similar mechanisms operate (if they are tied at least to one contributor). Individual network among those who did not participate in the wild cat strike. They parameters are effective because of proportional conformity. Pro- also seek reinforcement, and may form the group of “moderates” portional conformity benefits of a new tie are the highest for an or “rational egoists”. In a denser community, however, it is less individual with just one existing tie to a dissimilar actor. For this likely that such an event can contribute to the building of a larger or individual, the proportion of similar actors is improved by a half an even denser social network, because the relative improvements if a new tie is formed. Starting from more ties or from more ties workers can attain in terms of proportional conformity are small to similar actors mean less improvement in the composition and if they are related to many cooperators and many defectors at the hence less proportional conformity gains. same time. Nevertheless, those who have many abridging relations In a utopian setting in which contacts are formed freely, all may seek new acquaintances even in a dense network. contributors would be interested to be matched with all other con- Nonetheless, not only structural characteristics matter. Costs of tributors and all defectors would be happy to build relations with changing ties also have an impact on decisions. Large values of a other defectors to enjoy higher behavioral confirmation rewards. In and f can be interpreted as high constraints on network change case selective incentives are more important than behavioral con- imposed by the given network. For example, consider the case of firmation, contributors would even be interested to get any kind the strike in a project team whose members are tied by a network of connections also including ties to defectors. A symmetric rela- of such dyadic task interdependencies that are important for their tionship requires a mutual agreement of the parties, however, and future work performance and thus also their career prospects. In defectors would veto this, because additional cross group ties may such a situation, team members would not easily segregate along reduce the benefits they enjoy from proportional conformity. More- the lines of strikers versus goons, because other rewards besides over, in a full contribution strategy profile highest benefits would selective incentives and behavioral confirmation are at stake when come from a network in which everyone is tied to everyone else.11 relations change. Like for link deletion, network characteristics influence tie formation only within the subset of those members who have 4. Simultaneous social control and strongly robust network abridging ties. Within this group, strongly embedded, central actors equilibrium will be less likely to change their networks. Unlike in the case of In the previous section we relaxed the traditional assumption of models of collective action that the social network is given and indi-

10 Similar results are obtained also for negative social control (see Appendix B). viduals cannot change their relations. The analysis we provided is 11 Much less (if any) tie formation can be expected if social control is expressed in particular suitable for situations in which social control mecha- only as punishments (see Appendix B). nisms are delayed compared to participation decisions in collective 184 K. Takacs´ et al. / Social Networks 30 (2008) 177–189 action. There are situations, however, when structural changes and work equilibria is restricted to extreme configurations: only full behavior in the collective action game are simultaneous. Further- contribution and full defection with complete networks can be strongly more, even when this is not the case, actors can anticipate structural robust network equilibria. Part (b) states that when the group is changes at the time of their participation decision in collective small and building and deleting costs a and f are relatively small action. Under such circumstances, these actions are part of the compared to selective incentives and mass conformity a partial same strategy; structural decisions and network stability should contribution profile cannot be strongly robust network equilibrium be considered together with individual decisions and equilibria in (see Appendix A for the proof). collective action. For example, workers who participate in a wild Theorem III. cat strike may take into account the risk of loosing friends and the opportunity of finding new ones. Let us consider a worker, who works in a peripheral unit in which the majority does not prefer (a) If a =0, f =0 and (b1 >0 or s > 0), then in strongly robust network equilibrium (R, ␴*): R = R and ∗ = ∗ for all i,j ∈ N. to join the strike of the major workshop. If this worker has some N i j − ≥ friends in the major workshop, then she has to decide about partic- (b) If s +2b1 > nf +(n 2)a and b1 f, then in strongly robust network ␴* ∗ = ∗ ∈ 12 ipation but also about the community she wants to be embedded equilibrium (R, ): R = RN and i j for all i,j N. in at the same time. To address situations like this one, we need an equilibrium To see the intuition underlying this result, consider a network refinement that embraces the concepts of unilateral stability and in which there are both defectors and contributors. Without costs Nash equilibrium in the context of games played in social networks. for link changes, either contributors or defectors would profit from For this purpose, we propose the notion of strongly robust network changing their contribution decision, abandoning all ties to their equilibrium. A network of social relations and a strategy profile in ex-group and build connections to every member of their new the (collective action) game are in strongly robust network equi- group. Contributors are better off by becoming “integrated” defec- librium, if there is no actor, for whom any combination of changes tors if in the original network there are a sufficiently large number in her contribution decision and in her ego-network would result of defectors. In this case, the related gain in behavioral confirma- in a better outcome; requiring consent for every new relation from tion exceeds the loss in terms of foregone provision of the collective partners. good and lost selective incentives. Conversely, if the number of defectors falls below this critical level, then all defectors would gain Definition. We define the combination of R and strategy profile from turning into contributors and complete their network with ␴*(∗,∗ )astrongly robust network equilibrium, i −i fellow contributors. Hence, the network in which there are both defectors and contributors is not in equilibrium, because depend- ∗ ∗ ≥ ∗ ∪ \ − − if i((i ,−i), R) i((i, −i), (R Gi) Hi) hia gif ing on the proportion of contributors in the entire network either ∈ ∈ ∗ ∗ or exists such j N that ij Gi: j((i, −i), R)>j((i, −i), all defectors or all contributors are pulled towards an equilibrium R ∪ {ij}) − f with uniform choices. One should note that if forming and delet- ⊂ ˇ ⊂ ∈ for any Gi Ri,anyHi Ri and for all i N. ing ties are free or have relatively low costs, then the assumption of simultaneous decisions about relationships and participation Note that just as in unilateral stability, the concept of strongly makes the initial network structure irrelevant. When looking at robust network equilibrium requires that no deviations are individ- network changes only, the initial network structure matters as it ually beneficial given that strategies of others and relations in which constrains the possible sets of contributors and free riders. But the i is not involved are fixed. From a new partner j, consent is required initially given network can also be important for collective action only for the formation of ij. A new partner j would not give consent if behavioral changes are considered simultaneously with network if her subjective rewards in the network without ij were higher than updates. The key assumption needed for this is that there are costs her subjective rewards in the network with ij considering the new for deleting or building ties that represent structural constraints strategy profile, in which only the action of i might be different than embodied in the existing network. the original choice. This formalization is the straightforward way to Costly network changes imply that not only complete net- capture that strategy choices and network choices are simultaneous works can be strongly robust network equilibria. When a cost and not independent. of a new tie exceeds mass conformity benefits (f > b1), any What are the conditions under which strongly robust network initial network without isolates is in strongly robust network equilibria can occur in collective action? It follows immediately that equilibrium in a full defection profile. Furthermore, any initial net- only a Nash equilibrium strategy profile and only unilaterally stable work with a minimum individual degree of rmin is in strongly robust networks can be in strongly robust network equilibrium. However, network equilibrium in a full cooperation profile if the cost of a unilateral stability and Nash equilibrium in the collective action new tie exceeds selective incentives and mass conformity bene- game are necessary but not sufficient conditions for strongly robust fits (f > s + b1) and social control rewards exceed contribution costs network equilibrium. Consider for instance, a situation in which a (˛ + b2 + rmin(b1 + s)>c to assure that switches to defection are not disconnected network with a complete component of contributors beneficial). and a complete component of defectors is unilaterally stable. Given The exact condition for part (b) of Theorem III also determines sufficiently high benefits from receiving selective incentives and the cost constraint above which partial contribution with complete behavioral confirmation, the strategy profile can also be in Nash components of defectors and contributors can be a strongly robust equilibrium. In this situation, nobody has an incentive to aban- network equilibrium. In combination with the arguments above it don relations, to form ties and receive agreement from the new can be also stated that not only complete components can charac- partners, or to change the decision in the collective action game. terize partial contribution in strongly robust network equilibrium. Restructuring relations and changing the action in collective action, For not complete components of defectors f > b1 should hold, for not however, can be beneficial for some players. If forming new ties and abandoning existing relations are free, there would always be play- ers for whom such changes were beneficial. Part (a) of Theorem III 12 In case of negative social control, when deleting ties is free, only full defection expresses that when there are no costs of network change and social with complete isolation can be strongly robust network equilibrium (see Appendix control is expressed as rewards, the number of strongly robust net- B). K. Takacs´ et al. / Social Networks 30 (2008) 177–189 185 complete components of contributors f > s + b1 should be satisfied. enjoy more rewards of social control, individuals might strategically Besides, in equilibrium, every contributor including the one with revise their network relations. Interestingly, this does not imply the fewest connections (rcmin ) must be better off by remaining con- that networks that show at the outset a clear tendency towards + + + tributor than by switching to defection (˛ b2 rcmin (b1 s) >c) segregation of contributors and defectors may have the highest or by switching to defection and integrating with any number of potential for network segregation in the course of the collective + + + + − defectors (˛ rcmin (b1 s a) >c b1 f ; this condition is rele- action. We demonstrated in particular that if it is possible to delete vant if b2 is large). ties, then initial segregation will not be a predictor of final segre- Finally, if multiple conditions are met, tie formation and dele- gation. On the other hand, while initially sparser sub-networks of tion costs also make it possible that ties between contributors and contributors and defectors may be subject to link deletion, more defectors are preserved in strongly robust network equilibria.13 densely knit subgroups may be stable and preserve bridging ties between dissimilar actors. This can be the case even if the rela- 5. Discussion tive measures of segregation and degree variance are the same in the original networks compared. This is caused by the behavioral Previous research on social network effects in collective action assumption of proportional conformity. Proportional conformity has shown that networks may have a double edge. Sometimes, implies that a weakly integrated bridging actor who deletes bridg- social influence processes in networks may facilitate collective ing ties will achieve a larger improvement in the composition of action, but at other times social ties may be instruments for her relations than a well-integrated bridging actor who deletes the deviants to resist conformity pressure or to affirm each others’ same number of bridging ties. deviant behavior. This suggests that actors face not only incentives Another model implication is that denser networks are less to contribute or defect, but also incentives to make or break social likely to be subject to link deletion as well as to tie formation relations that are interdependent with their contribution decisions. than sparser networks. Furthermore, individual level analysis of the Hence, an analysis of the conditions under which networks promote possibility of forming new ties demonstrated that in partial con- collective action cannot be disentangled from an analysis of the tribution those actors can particularly improve their situation by dynamics of network relations between prospective participants. connecting to others with similar choices who have only few con- Previous models of collective action have considered social rela- nections and relatively many of them lead to dissimilar actors. This tions as given. Individuals change their network ties, however, and means that there is an effect of initial segregation on tie formation: changes are partly consequences of collective interdependences. less segregated networks that contain actors who have several ties This paper investigated the stability of collective action and net- to dissimilar others are more likely to be exposed to tie formation work structures that are subject to endogenous changes. than more segregated ones. The impact of segregation on tie for- We proposed an integrated game theoretic model and derived mation, however, is not as strong as the effect of individual degree: equilibrium predictions without the need of using simulation new ties are most likely formed between two similar actors who methods. In the proposed model, network effects were incorpo- have only few ties. rated in the standard n-person public goods game through different For a synthesis of analyses, we introduced a new equilibrium social control mechanisms that assume social rewards are transmit- concept of strongly robust network equilibrium that combines ted through interpersonal relations. Forms of social control, namely equilibrium in collective action and network stability. The use of selective incentives and forms of behavioral confirmation were this concept is not restricted to the structurally embedded public modeled as rewards (and punishments) that influence individual good game. It can be applied for the analysis of any non-cooperative decisions through actors’ relationships to relevant others and make game in which payoffs are partly dependent on decisions about collective action possible. Social control that goes together with col- interpersonal relations that are defined within the set of players. lective action may also lead actors to break relations in the network Real-word interactions often involve simultaneous decisions about and build new ones. In order to avoid unpleasant influence and to actions and connections that influence the payoffs of those actions (see Bramoulle´ et al., 2004; Goyal and Vega-Redondo, 2005). Col- lective action is just an example at hand. Game theoretical analysis of any such situation could be helped by the adoption of a concept 13 First, deletion costs are sufficiently high such that the defector has no interest to like strongly robust network equilibrium. delete this tie (a > b2/(rid + 1)). Second, the defector has no interest to change to con- tribution with any combination of network update. This means that the subjective Our equilibrium concept applied to collective action has yielded payoffs ridb1 + rid b2/(rid +1)+c should be larger than some interesting implications. We have shown that in the ideal typical case where there are no costs for structural change, only

(a) the subjective payoffs from a pure switch ˛ + s + b1 + b2/(rid + 1); full contribution and full defection with complete networks can be

(b) the subjective payoffs from a switch and deletions ˛ + s + b1 + b2 − rida; strongly robust network equilibria assuming positive social control.

(c) the subjective payoffs from a switch and integration ˛ + rics + ricb1 + In case of negative social control and no costs of deleting relations, − ricb2/(rid + ric) ricf; full defection and isolation is the only strongly robust network equi-

(d) the subjective payoffs from a switch, deletions and integration ˛ + rics + ricb1 + librium. For the more realistic case where network change is costly, − − b2 rida ricf. we could show that network structures that integrate contributors and defectors can be stable. Hence, the assumption that actors in Third, the contributor is better off by staying at contribution collective action make purposive decisions about both their contri- than by switching to defection in combination with any update butions and their social relations does neither imply an inexorable of her network. This means that the subjective payoffs ricb1 +(ric +1)s + ric b2/(ric +1)+˛ should be larger than tendency towards “deviant cliques”, nor does it suggest that uni- versal contribution or universal defection are likely outcomes in

(a) the subjective payoffs from a pure switch c + b1 + b2/(ric + 1); collective actions that are embedded in existing networks.

(b) the subjective payoffs from a switch and deletions c + b1 + b2 − rica; Our aim was to provide a foundation for subsequent research

(c) the subjective payoffs from a switch and integration c + ridb1 + rid that recognizes the interrelation of collective action and network b2/(rid + ric) − ridf; structure. The presented framework can be extended to similar

(d) the subjective payoffs from a switch, deletions and integration c + ridb1 + b2 situations with different collective structure of interdependence, − − rica ridf. including public good provision with a different production func- 186 K. Takacs´ et al. / Social Networks 30 (2008) 177–189 tion, sustaining a public bad (cf. Kuran, 1995) and other n-person ment is larger than the cost a of deleting a tie, then i(␴, games. Another assumption that can be relaxed in future research R\{ij}) − a > i(␴, R). is the binary character of social relations (two individuals are Note that ij could also be not stable even if this latter condition either tied or not). An alternative approach would be to assume is not met as it might be an element of a set Hi of links to be weighted ties that model differences between, e.g., good friends and deleted that increases the subjective reward of the defector. The mere acquaintances in the network. In such a framework, ties are number of elements in Hi is hi (ri ≥ ric ≥ hi ≥ 0), where individual not abandoned or built, but weights are reconsidered. This model i should choose hi as to maximize the net benefits of her action. extension would also allow considering asymmetric ties (for a sim- This means that we should find the maximum value of ilar dynamic analysis see Kitts et al., 1999). hirid − We adopted a model based on strategic individual actions b2 hia, (7) r (r − h ) about cooperation and network relations. Strategic decision mak- i i i ing in a situation where both network relations and contributions which is the difference between the benefits and the costs of decisions might change may impose an implausibly high cogni- abandoning hi relations to contributors. As hi increases, the net tive load on decision makers. Possible ways of relaxing the strict benefits are also increasing (if rid > 0). Hence, the optimal strat- cognitive assumptions of the model is to incorporate boundedly egy for a defector who has at least one connection to another rational learning in repeated decision making, such as reinforce- defector is to delete all her ties to contributors, which holds also ment learning (cf. Macy, 1993; Erev and Roth, 1998; Macy and for the case of negative social control (see later in a Appendix Flache, 2002), or myopically forward-looking belief learning (or B). When the benefits of deleting the first connection outweigh “fictitious play”, cf. Fudenberg and Levine, 1998), or modify the the costs a, deleting hi > 1 connections will outweigh hia with a model with empirically justified assumptions on behavior and larger margin. Hence it might happen that deleting one or few available information. Another further step would be an analysis connections does not outweigh the costs, but deleting all links of strategic behavior in collective action with sequential indi- to contributors becomes beneficial. In short, relation ij is not vidual decisions both in the repeated public goods game and in stable, if at least one of the actors (i) benefits from deleting all the repeated network game. Consequently, a dynamic interre- of her ric connections. After substituting to (3) it means that ij lated analysis of repeated collective action problems and structural is not stable,if dynamics that might explain self-reinforcing spirals of participa- b2 tion could also be performed. For the complexity of this problem, >a (8) r however, agent-based simulation techniques would be more appro- i priate than analytical methods. These proposed developments are holds, given that rid >0. purely theoretical. The potential of our approach to also inform (c) If rid = 0, deleting ij might be beneficial for i in combination with empirical research is demonstrated by a recent study that pro- building some new ties to defectors. This is the case when the poses laboratory experiments with artificial networks to test some costs of change are compensated by behavioral confirmation of the model predictions outlined in this paper (Takacs´ and Janky, benefits. As marginal gains are highest for the first newly 2007). formed tie, ij is not stable and a new tie to another defector is formed, if b1 + b2/ric > a + f. Note that deleting a larger subset of Acknowledgements links to contributors here does not provide higher subjective rewards for i than just deleting ij and forming a new relation to a We would like to thank Arnout van de Rijt, Marcel van Assen, defector. On the other hand, deleting all ric links to contributors Russell Hardin, Anthony Oberschall, Zoltan´ Szant´ o,´ Benedek Kovacs,´ and forming a new tie to a defector is beneficial if b1 + b2 > rica + f, Christian Steglich, Attila Gulyas,´ Peter´ Csoka,´ and three reviewers which provides the same marginal benefits as just of Social Networks for their comments on an earlier version of the deleting ij. paper. The authors acknowledge support of the Hungarian Scientific Research Fund (OTKA), T/16, 046381; the Netherlands Organization Proof of Theorem II. for Scientific Research (NWO) VIDI scheme (452–04–351); Col- legium Budapest/Institute for Advanced Study; and the Netherlands (a) Follows immediately as there are no gains for the defector. Institute for Advanced Study in the Humanities and Social Sciences (b) Consider a contributor i who has the option to form multiple (NIAS). ties to other cooperators at the same time. The cost of form- ing one tie (f) is the same for every new relation and for both Appendix A sides. Denoting the number of new ties of individual i by gi, the benefits of structural change for i are given by r s + b + b id − f. (9) Proof of Theorem I. 1 2 + ri(ri gi)

(a) If i = j = 0, then by deleting ij, i decreases by a non-negative As gi increases, the marginal benefits are decreasing (if rid > 0). proportion of behavioral confirmation and there is no source Hence, the highest marginal benefits for a contributor come of compensation. Even when behavioral confirmation incen- from a first new tie to another contributor. tives are zero, there is no improvement by breaking ij.If (c) Substituting 1 to gi in Eq. (5) and expressing f, provides the i = j = 1, then by deleting ij, i decreases by some proportion condition. of behavioral confirmation and by s and there is no source of (d) The result for defectors is obtained the same way as for con- compensation. tributors. (b) The payoff of defector i in any strategy profile ␴ can consist of only the following elements: c, nc˛, ridb1, and ridb2/ri. The Proof of Theorem III. For the proof of parts (a) and (b) of the first three elements do not change if she abandons a relation theorem, we demonstrate that in partial contribution equilibria to a contributor. The last element has an increment of ridb2/ri with nc contributors and nd defectors (nc + nd = n), either contrib- − (ri 1), which is always positive if b2 > 0 and rid > 0. If this incre- utors or free riders would have the incentive to abandon all their K. Takacs´ et al. / Social Networks 30 (2008) 177–189 187 existing ties, form new ties with every member of the other camp a free rider might have an incentive to delete a tie to another free and change their action in the collective action game. Contributors rider only once all ties to contributors have already been deleted. are better off by remaining in the complete component of contrib- When this is the case, a free rider will delete all relations, if s > a utors than by switching to defection and connect to all defectors holds; and will delete no relations to other free riders, if s < a. if Part (c) states that contributors prefer to delete their links to defectors if deletion costs are under a threshold determined by ˛ + (n − 1)s + (n − n − 1)b ≥ c − (n − 1)a − n f. (10) c c d 1 c d behavioral confirmation and individual degree of the contributor. On the other hand, defectors are better off by remaining in the A contributor i prefers to delete a link with a defector, if complete component of defectors than switching to contribution r b + b ic >a (7c-) and connect to all contributors, if 1 2 − ri(ri 1) + + − + ≤ + − + ˛ ncs (nc nd 1)b1 c (nd 1)a ncf. (11) holds. A contributor prefers to delete all links with defectors, if Eqs. (10) and (11) cannot be simultaneously satisfied when a =0, f = 0, and b1 >0 or s > 0, which completes the proof of part b2 - b1 + >a. (8c ) (a). These equations cannot be simultaneously satisfied also when ri s +2b > nf +(n − 2)a. If there are costs of tie formation, for the con- 1 Hence, the most beneficial for a contributor is to delete all her sent of defectors it is required that b ≥ f, which provide the cost 1 ties to defectors. conditions for part (b). - Corollary I.a . Any R is strong link deletion proof if i =1for all i ∈ N. Appendix B Corollary I.b-. If s > a, then R is strong link deletion proof if there are links only between contributors. Main results in case social control appears in form of punishments. We assume in the following calculations that selective incen- In general, in case of negative social control, the network will be tives and behavioral confirmation are punishments that decrease strong link deletion proof in a strategy profile, if there is no defector individual payoff. These calculations can be contrasted with those for whom (8d-) holds, and there is no contributor for whom (8c-) in the main text that hold for positive rewards of social control. The holds. subjective payoffs of defection and contribution for i are now: Theorem II. In any R and given ␴, for all i,j ∈ N and ij ∈/ R: ␴ = + + − + b2 − − − i( , R) ˛ ris (ric rid) b1 c i ris ricb1 ri (a) If i =0and j =1,then i(␴, R) ≥ i(␴, R ∪ {ij}). ␴ ≥ ␴ ∪ { } − ␴ ≥ ␴ n (b) If i = j and i( , R) i( , R ij ) f, then i( , R) i( , r − ic + - R ∪ G ) − g f, where ij ∈ G for any G ⊂ Rˇ . b2 ˛ j, (1 ) i i i i i ri (c) If = =1,b r /(r2 + r ) >f, then ij is initiated by i. j=1 i j 2 id i i 2 + + (d) If i = j =0,b2ric/(r ri) >f s, then ij is initiated by i. where j ∈ N\{i}. From these equations it follows that the conditions i for participation of individual i to be beneficial are exactly the same Proof. Part (a) follows directly from the payoffs. For other parts as in Eq. (2). of Theorem II, consider that contributor i has an incentive to be Theorem I-. In any R and given ␴, for all i,j ∈ N and ij ∈ R: matched with contributor j, assuming a cost of forming a tie f,if r b id >f, ␴ ≥ ␴ \{ } 2 + (a) If i = j =1,then i( , R) i( , R ij ). ri(ri 1) (b) If =0and s + b + (b /r ) >a, then: i 1 2 i which follows from a comparison of marginal benefits and costs. if =1,then ij is not stable. j Defector i has an incentive to be matched with defector j, assuming if j =0and s > a, then ij is not stable. + a cost of forming a tie f,if (c) If i =1,j =0,and b1 (b2/ri) >a, then ij is not stable. r b ic >f+ s. 2 + Proof. Part (a) states that a contributor always prefers to keep the ri(ri 1) links with other contributors. This follows from Eq. (1-). Part (b) states that if deletion costs are under a threshold Consider now that contributors can form multiple ties at the determined by selective incentives, behavioral confirmation, and same time. The cost of forming one tie f is the same for every new individual degree of the defector, then the defector i prefers to delete relation and for both sides. Denoting the number of new ties of the ij link to the contributor. She prefers to delete a link with a contributor i by gi, this structural change is beneficial for i,if contributor, if r b id >f (9-) r 2 r (r + g ) s + b + b id >a. (7d-) i i i 1 2 − ri(ri 1) is satisfied. This shows that the marginal benefits of forming more She prefers to sever all links with contributors, if ties are decreasing. Similarly, the marginal benefits of forming new ties are decreasing also for defectors. b s + b + 2 >a. (8d-) As it was discussed, when s > a holds, a defector prefers to delete 1 r i all of her relations. When s < a and (8d-) hold, a defector deletes The left side of Eq. (8d-) is never smaller than the left side of all relations to contributors and builds no ties. When (8d-) is not Eq. (7d-), hence the most beneficial for i is to delete all her ties to satisfied, but (9-) holds (high costs of link deletion and cheap tie contributors. formation), then defectors are interested to build at least one tie to When behavioral confirmation has an effect, deleting a tie to another defector, but keep all their ties to contributors. a contributor always gives higher benefits for the free rider than A similar result can be obtained for contributors. When (8c-) deleting a tie to another free rider. This has the consequence that holds, a contributor prefers to delete all ties to free riders and has no 188 K. Takacs´ et al. / Social Networks 30 (2008) 177–189 interest to form new ties. When (8c-) is not satisfied and (9-) holds, Dutta, B., Jackson, M.O., 2003. On the formation of networks and groups. In: Dutta, B., a contributor would like to form a new tie to another contributor, Jackson, M.O. (Eds.), Networks and Groups. Models of the Strategic Formation. Springer-Verlag, Berlin. but is not interested to delete any of her ties. Dutta, B., Mutuswami, S., 1997. Stable networks. Journal of Economic Theory 76, Hence, a combination of deleting and forming ties is never prof- 322–344. itable in case of negative social control. This has the corollary that Dutta, B., van den Nouweland, A., Tijs, S., 1998. Link formation in cooperative situa- tions. International Journal of Game Theory 27, 245–256. a strong link deletion proof network in which no dyad is interested Erev, I., Roth, A.E., 1998. Predicting how people play games: reinforcement learning in to form a new connection is resistant to changes (and unilaterally experimental games with unique, mixed strategy equilibria. American Economic stable). Review 88, 848–879. Fehr, E., Gachter,¨ S., 2000. Cooperation and punishment in public good experiments. Theorem III. If a =0and s >0,then in strongly robust network equi- American Economic Review 90, 980–994. Fehr, E., Gachter,¨ S., 2002. Altruistic punishment in humans. Nature 415 (January), * * librium (R, ␴ ): R is an empty set and ␴ = 0. 137–140. Flache, A., 1996. The Double Edge of Networks. An Analysis of the Effect of Informal Theorem III states when deleting relations is free, only full Networks on Cooperation in Social Dilemmas. Thesis Publishers, Amsterdam. defection with no ties can be strongly robust network equilibrium, Flache, A., 2002. The Rational Weakness of Strong Ties. Failure of Group Solidarity in a Highly Cohesive Group of Rational Agents. Journal of Mathematical Sociology irrespective of the costs of forming new ties. This follows directly 26, 189–216. from (1-): as there are no positive rewards for maintaining rela- Flache, A., Macy, M.W., 1996. The Weakness of Strong Ties: Collective Action Failure tions and tie deletion is free, all defectors are better off by deleting in a Highly Cohesive Group. Journal of Mathematical Sociology 21, 3–28. Fudenberg, D., Levine, D., 1998. The Theory of Learning in Games. MIT Press, Boston. all their relations by at least s and contributors are gaining at least Gilles, R.P., Sarangi, S., 2004. The Role of Trust in Costly Network Formation. Virginia c − ˛ by switching to defection and deleting all ties. (Cooperation Tech, Department of Economics Working Paper. http://nw08.american.edu/ can only be maintained in strongly robust network equilibrium if ∼hertz/Spring%202004/NetworkTrust3.pdf. Gould, R.V., 1993a. Collective action and network structure. American Sociological tie deletion costs are high compared to costs of cooperation and Review 58, 182–196. punishments of social control are serious.) Gould, R.V., 1993b. Trade cohesion, class unity and urban insurrection: artisanal activism in the Paris commune. American Journal of Sociology 98, 721–754. Gould, R.V., 2003. Why do networks matter? Rationalist and structuralist inter- pretations. In: Diani, M., McAdam, D. (Eds.), Social Movements and Networks. References Relational Approaches to Collective Action. Oxford University Press, Oxford/New York. Bala, V., Goyal, S., 2000. A noncooperative model of network formation. Economet- Goyal, S., Vega-Redondo, F., 2005. Network formation and social coordination. Games rica 68, 1181–1229. and Economic Behavior 50, 178–207. Belleflamme, P., Bloch, F., 2004. Market sharing agreements and collusive networks. Goyal, S., Vega-Redondo, F., 2007. Structural holes in social networks. Journal of International Economic Review 45, 387–411. Economic Theory 137, 460–492. Berninghaus, S.K., Schwalbe, U., 1996. Evolution, interaction, and Nash equilibria. Harsanyi, J.C., 1969. Rational choice models of political behavior vs. functionalist and Journal of Economic Behavior and Organization 29, 57–85. conformist theories. World Politics 21, 513–538. Blum, R.W., Beuhring, T., Rinehart, P.M., 2000. Protecting teens: Beyond race, income Haynie, D., 2001. Delinquent peers revisited: does network structure matter? Amer- and family structure. Center for Adolescent Health, University of Minnesota, ican Journal of Sociology 106 (4), 1013–1057. Minneapolis, MN. Hechter, M., 1987.Principles of Group Solidarity. University of California Press, Berke- Bonacich, P., 2001. Structural holes in structural wholes: a simulation of individ- ley. ual strategy and organizational form. In: Paper Presented at the 2001 Annual Heckathorn, D.D., 1988. Collective Sanctions and the Creation of Prisoner’s Dilemma Meeting of the American Public Choice Society. Norms. American Journal of Sociology 94 (November (3)), 535–562. Boyd, R., Richerson, P.J., 1985. Culture and the Evolutionary Process. Chicago Univer- Heckathorn, D.D., 1989. Collective action and the second-order free-rider problem. sity Press, Chicago. Rationality and Society 1, 78–100. Boyd, R., Gintis, H., Bowles, S., Richerson, P.J., 2003. The evolution of altruistic punish- Heckathorn, D.D., 1996. The dynamics and dilemmas of collective action. American ment. Proceedings of the National Academy of Sciences of the U.S.A. 100 (March Sociological Review 61, 250–277. (18)), 3531–3535. Hirshleifer, D., Rasmusen, E., 1989. Cooperation in a repeated prisoners’ Bramoulle,´ Y., Lopez-Pintado,´ D., Goyal, S., Vega-Redondo, F., 2004. Network for- dilemma with ostracism. Journal of Economic Behavior and Organization 12, mation and anti-coordination games. International Journal of Game Theory 33, 87–106. 1–19. Jackson, M.O., 2004. A survey of models of network formation: stability and effi- Burger, M., Buskens, V., 2006. Social context and network formation: experimental ciency. In: Demange, G., Wooders, M. (Eds.), Group Formation in Economics: studies. ISCORE Paper, 234, Utrecht University, Netherlands. Networks, Clubs and Coalitions. Cambridge University Press, Cambridge. Burt, R.S., 1992. Structural Holes: The Social Structure of Competition. Harvard Uni- Jackson, M.O., van den Nouweland, A., 2005. Strongly stable networks. Games and versity Press, Cambridge (Mass.). Economic Behavior 51, 420–444. Burt, R.S., 2005. Brokerage and Closure. An Introduction to Social Capital. Oxford Jackson, M.O., Watts, A., 2002a. On the formation of interaction networks in social University Press, Oxford. coordination games. Games and Economic Behavior 41, 265–291. Buskens, V., Snijders, C., 2005. Effects of network characteristics on reaching the Jackson, M.O., Watts, A., 2002b. The evolution of social and economic networks. payoff-dominant equilibrium in coordination games: a simulation study. ISCORE Journal of Economic Theory 106, 265–295. paper, 232. Utrecht University, Netherlands. Jackson, M.O., Wolinsky, A., 1996. A strategic model of social and economic networks. Buskens, V., van de Rijt, A., 2005. Dynamics of networks if everyone strives for Journal of Economic Theory 71, 44–74. structural holes. ISCORE Paper, 227. Utrecht University, Netherlands. Janky, B., Takacs,´ K., 2005. Social control, network structure, and participation in Chong, D., 1991. Collective Action and the Civil Rights Movement. Chicago University collective action. In: Bodo,´ B., et al. (Eds.), Tarsadalmi´ terben´ [In Social Space]. Press, Chicago. BME, Budapest, pp. 157–188. Chwe, M.S.-Y., 1999. Structure and strategy in collective action. American Journal of Kim, H., Bearman, P.S., 1997. The structure and dynamics of movement participation. Sociology 105, 128–156. American Sociological Review 62, 70–93. Chwe, M.S.-Y., 2000. Communication and coordination in social networks. Review Kitts, J.A., 2006. Collective action, rival incentives, and the emergence of antisocial of Economic Studies 67, 1–16. norms. American Sociological Review 71, 235–259. Coleman, J.S., 1990. The Foundations of Social Theory. The Belknap Press, Cambridge, Kitts, J.A., Macy, M.W., Flache, A., 1999. Structural learning: attraction and conformity MA. in task-oriented groups. Computational and Mathematical Organization Theory Diani, M., 2003a. Introduction: social movements, contentious actions, and social 5 (2), 129–145. networks: ‘From Metaphor to Substance’? In: Diani, M., McAdam, D. (Eds.), Social Kuran, T., 1995. Private Truths, Public Lies: The Social Consequences of Preference Movements and Networks. Relational Approaches to Collective Action. Oxford Falsification. Harvard University Press, Cambridge (Mass.). University Press, Oxford/New York. Lovaglia, M.J., Willer, R., Troyer, L., 2003. Power, status, and collective action: devel- Diani, M., 2003b. Networks and social movements: a research programme. In: Diani, oping fundamental theories to address a substantive problem. In: Thye, S.R., M., McAdam, D. (Eds.), Social Movements and Networks. Relational Approaches Skvoretz, J., Lawler, E.J. (Eds.), Advances in Group Processes, vol. 20. JAI Press, to Collective Action. Oxford University Press, Oxford/New York. Greenwich, CT. Diani, M., 2003c. ‘Leaders’ or brokers? Positions and influence in social movement Macy, M.W., 1991. Chains of cooperation: threshold effects in collective action. Amer- networks. In: Diani, M., McAdam, D. (Eds.), Social Movements and Networks. ican Sociological Review 56, 730–747. Relational Approaches to Collective Action. Oxford University Press, Oxford/New Macy, M.W., 1993. Backward looking social control. American Sociological Review York. 58, 819–836. Dixon, M., Roscigno, V.J., 2003. Status, networks, and social movement participation: Macy, M.W., Flache, A., 2002. Learning dynamics in social dilemmas. Proceedings of the case of striking workers. American Journal of Sociology 108, 1292–1327. the National Academy of Sciences 99 (May (Suppl. 3)), 7229–7236. K. Takacs´ et al. / Social Networks 30 (2008) 177–189 189

Macy, M.W., Kitts, J., Flache, A., Benard, S., 2003. Polarization in dynamic networks: Forgas, J.P., von Hippel, W. (Eds.), The Social Outcast: Ostracism, Social Exclusion, a Hopfield model of emergent structure. In: Breiger, R., Carley, K., Pattison, P. Rejection, and Bullying. Psychology Press, New York. (Eds.), Dynamic Social Network Modeling and Analysis. Workshop Summary Ploeger, M., 1997. Youth employment and delinquency: reconsidering a problematic and Papers. National Academy Press, Washington, DC, pp. 162–173. relationship. Criminology 35 (4), 659–675. Marwell, G., Oliver, P.E., 1993. Critical Mass in Collective Action. Cambridge Univer- Sandell, R., Stern, C., 1998. Group size and the logic of collective action: a network sity Press, Cambridge. analysis of a Swedish Temperance Movement 1896–1937.Rationality and Society Marwell, G., Oliver, P.E.,Prahl, R., 1988. Social networks and collective action: a theory 10 (3), 327–345. of the critical mass. III. American Journal of Sociology 94, 502–534. Simons-Morton, B., Chen, R.S., 2006. Over time relationships between early adoles- McAdam, D., 1986. Recruitment to high-risk activism: the case of freedom summer. cent and peer substance use. Addictive Behaviors 31, 1211–1223. American Journal of Sociology 92, 64–90. Slikker, M., van den Nouweland, A., 2001. A one-stage model of link formation and McAdam, D., Paulsen, R., 1993. Specifying the relationship between social ties and payoff division. Games and Economic Behavior 34, 153–175. activism. American Journal of Sociology 993, 640–667. Snijders, T.A.B., Baerveldt, C., 2003. A multilevel network study of the effects of delin- Morris, S., 2000. Contagion. Review of Economic Studies 67, 57–78. quent behavior on friendship evolution. Journal of Mathematical Sociology 27, Oberschall, A., 1973. Social Conflict and Social Movements. Prentice-Hall, Englewood 123–151. Cliffs (NJ). Takacs,´ K., 2001. Structural embeddedness and intergroup conflict. Journal of Conflict Oberschall, A., 1993. Social Movements: Ideologies, Interests, and Identities. Trans- Resolution 45, 743–769. action, New Brunswick, N.J. Takacs,´ K., Janky, B., 2007. Smiling contributions: social control in a public goods Oberschall, A., 1994. Rational choice in collective protests. Rationality and Society 6, game with network decline. Physica A 378, 76–82. 79–100. Tilly, C., 1978. From Mobilization to Revolution. Addison Wesley, Reading Oliver, P., 1980. Rewards and punishments as social approval for collective action: (Mass.). theoretical investigations. American Journal of Sociology 85, 1356–1375. Tilly, C., Wood, L.J., 2003. Contentious connections in Great Britain, 1828–1834. Oliver, P.E., 1984. “If You Don’t Do It, Nobody Else Will”: active and token contributors In: Diani, M., McAdam, D. (Eds.), Social Movements and Networks. Relational to local collective action. American Sociological Review 49, 601–610. Approaches to Collective Action. Oxford University Press, Oxford/New York. Olson Jr., M., 1982. The Rise and Decline of Nations: Economic Growth, Stagnation, Ule, A., 2005. Exclusion and Cooperation in Networks. Thela Thesis. Tinbergen Insti- and Social Rigidities. Yale University Press, New Haven (CT). tute, Amsterdam. Opp, K., Gern, C., 1993. Dissident groups, personal networks and spontaneous co- van de Rijt, A., Buskens, V., 2005. A Nash Equilibrium Refinement for Myerson’s operation: the East German revolution of 1989. American Sociological Review Network Formation Game. ISCORE Paper, 230. Utrecht University, Netherlands. 58, 659–680. Watts, A., 2001. A dynamic model of network formation. Games and Economic Osa, M., 2003. Networks in opposition: linking organizations through activists in the Behavior 34, 331–341. Polish People’s Republic. In: Diani, M., McAdam, D. (Eds.), Social Movements and Williams, K.D., Forgas, J.P., von Hippel, W., 2005. The social outcast: intro- Networks. Relational Approaches to Collective Action. Oxford University Press, duction. In: Williams, K.D., Forgas, J.P., von Hippel, W. (Eds.), The Social Oxford/New York. Outcast: Ostracism, Social Exclusion, Rejection, and Bullying. Psychology Press, Ouwerkerk, J.W., van Lange, P.A.M., Galucci, M., Kerr, N.L., 2005. Avoiding the social New York. death penalty: ostracism and cooperation in social dilemmas. In: Williams, K.D.,